The twentieth Fighter crew joined the eighth Air strength Command in Dec of '43, flying the P-38 in lengthy variety bomber escort function. the gang later switched over to the P-51 in July of '44. the gang destroyed a complete of 449 enemy airplane in the course of its strive against travel. Over one hundred fifty photographs, eight pages of colour, eighty pages.

Let (M, ω, Ψ) be a Hamiltonian G-manifold. Let π : P → M be a pre-quantization circle bundle. This means that P is a G-equivariant principal circle bundle with a connection one-form Θ, such that dΘ = −π ∗ ω, and such that the moment map Ψ is given by π ∗ Ψξ = Θ(ξP ) for all ξ ∈ g. ) Then the pullback π ∗ Ψ : P → g∗ is an exact moment map. 13. A linear combination of abstract moment maps is an abstract moment map. 14. If Ψ0 and Ψ1 are exact moment maps, then so is (1−ρ)Ψ0 +ρΨ1 for any smooth function ρ : M → [0, 1].

2) is called exact (cf. 8). A compatible two-form is then given by ω = −dµ. An exact moment map has the property that ΨH vanishes on M H for all H ⊆ G. 27. Many “classical” moment maps, such as the canonical moment map on a cotangent bundle, or the moment map on a pre-quantization circle bundle, are exact. 12. The following two examples exhibit functoriality properties of abstract moment maps with respect to G and M , respectively. 9. Let M be a G-manifold with an abstract moment map Ψ : M → g∗ .